Subgroups — #226404

Order 512.

632 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22722	C₂	1	2	yes	maximal
2	C₄×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	#9604	C₂	1	4	yes	maximal
4	#9601	C₂	1	2	yes	maximal
5	#8806	C₂	1	2	yes	maximal
6	#8806	C₂	1	1	yes	maximal, characteristic
7	#8274	C₂	1	2	yes	maximal
8	C₄×#335	C₂	1	1	yes	maximal, characteristic
9	C₂×#446	—	2	2	no	—
10	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
11	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
12	C₄²×Q₈	C₂²	1	1	yes	characteristic
13	#1940	—	2	4	no	—
14	#1939	—	2	2	no	—
15	#1842	C₂²	1	2	yes	—
16	#1842	C₂²	1	4	yes	—
17	C₈×C₄⋊C₄	C₄	1	2	yes	—
18	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
19	C₈×C₄⋊C₄	C₄	1	2	yes	—
20	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
21	#1767	C₂²	1	4	yes	—
22	C₄×C₄⋊C₈	C₂²	1	2	yes	—
23	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₈	C₂²	1	2	yes	—
25	C₄²×C₈	C₂²	1	1	yes	characteristic
26	C₄²×C₈	C₂²	1	2	yes	—
27	C₂×#335	C₂²	1	1	yes	characteristic
28	C₂×#335	C₂²	1	2	yes	—
29	C₂×#335	C₂²	1	4	yes	—
30	C₂×#335	C₂²	1	2	yes	—
31	C₂×#335	C₂²	1	1	yes	characteristic
32	#1277	C₂²	1	2	yes	—
33	#1277	C₂²	1	4	yes	—
34	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
36	C₂×C₄×Q₈	Q₈	1	1	yes	characteristic
37	#446	—	4	8	no	—
38	C₈×Q₈	—	2	4	no	—
39	C₈×Q₈	—	2	4	no	—
40	C₂×C₄⋊C₈	—	2	2	no	—
41	C₂×C₄⋊C₈	—	2	4	no	—
42	C₂×C₄⋊C₈	—	2	4	no	—
43	C₂×C₄⋊C₈	—	2	2	no	—
44	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₈	C₂³	1	2	yes	—
49	C₂×C₄⋊C₈	C₂³	1	4	yes	—
50	C₂×C₄⋊C₈	C₂³	1	2	yes	—
51	C₂×#97	—	2	4	no	—
52	C₂×C₄×C₈	—	2	2	no	—
53	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
54	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
58	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
59	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂³	1	2	yes	—
62	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
65	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
67	C₄×C₄⋊C₄	D₄	1	2	yes	—
68	C₄×C₄⋊C₄	Q₈	1	1	yes	characteristic
69	C₄×C₄⋊C₄	Q₈	1	2	yes	—
70	C₄³	C₂³	1	1	yes	characteristic
71	C₄³	C₂³	1	2	yes	—
72	#336	—	2	4	no	—
73	#336	—	2	2	no	—
74	#336	C₂×C₄	1	2	yes	—
75	#336	C₂×C₄	1	2	yes	—
76	#336	C₂×C₄	1	2	yes	—
77	#335	—	2	2	no	—
78	#335	—	2	4	no	—
79	#335	—	4	4	no	—
80	#335	—	4	2	no	—
81	#335	Q₈	1	4	yes	complemented
82	C₂²×Q₈	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₈	—	2	2	no	—
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	—	2	4	no	—
86	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₈	C₂²×C₄	1	2	yes	—
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂²×C₄	1	2	yes	—
90	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₄×Q₈	—	2	4	no	—
93	C₄×Q₈	—	2	2	no	—
94	C₄×Q₈	—	2	2	no	—
95	C₄×Q₈	D₈	1	4	yes	—
96	C₄×Q₈	Q₁₆	1	4	yes	—
97	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
99	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
100	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
101	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄²	C₂²×C₄	1	2	yes	—
104	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄²	C₂²×C₄	1	2	yes	—
106	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄²	C₂×D₄	1	2	yes	—
109	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄²	C₂×D₄	1	2	yes	—
111	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
112	C₂×C₄²	C₂×Q₈	1	2	yes	—
113	C₂×C₄²	C₂×Q₈	1	2	yes	—
114	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
115	C₂×C₄²	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₈	—	2	4	no	—
117	C₄⋊C₈	—	2	4	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	4	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	4	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	4	4	no	—
125	C₄⋊C₈	—	4	8	no	—
126	C₄⋊C₈	—	8	4	no	—
127	C₄⋊C₈	—	8	2	no	—
128	C₄⋊C₈	M₁₆	1	4	yes	—
129	C₄⋊C₈	C₂×Q₈	1	4	yes	—
130	#97	—	4	8	no	—
131	C₄×C₈	—	2	4	no	—
132	C₄×C₈	—	2	4	no	—
133	C₄×C₈	—	2	4	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	2	2	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	2	1	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	—	4	1	no	—
141	C₄×C₈	—	4	2	no	—
142	C₄×C₈	—	4	4	no	—
143	C₄×C₈	M₁₆	1	4	yes	—
144	C₄×C₈	C₂×Q₈	1	2	yes	—
145	C₄×C₈	#41	1	2	yes	—
146	#95	C₄⋊C₄	1	2	yes	—
147	#95	C₄⋊C₄	1	2	yes	—
148	#95	C₄⋊C₄	1	2	yes	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	2	2	no	—
151	C₂×Q₈	#107	1	4	yes	—
152	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
153	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
157	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
158	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
159	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
162	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	4	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	4	no	—
184	C₂×C₈	C₄×Q₈	1	2	yes	—
185	C₂×C₈	C₄×Q₈	1	2	yes	—
186	C₂×C₈	C₂×M₁₆	1	4	yes	—
187	C₂×C₈	C₂×M₁₆	1	2	yes	—
188	C₂×C₈	C₂×M₁₆	1	2	yes	—
189	C₂×C₈	#131	1	2	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	#107	1	8	yes	—
194	C₄⋊C₄	#107	1	4	yes	—
195	C₄⋊C₄	#108	1	4	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	4	1	no	—
209	C₄²	—	4	2	no	—
210	C₄²	C₂×M₁₆	1	2	yes	—
211	C₄²	C₂×M₁₆	1	2	yes	—
212	C₄²	C₂×D₈	1	2	yes	—
213	C₄²	C₂×D₈	1	4	yes	—
214	C₄²	C₂×Q₁₆	1	4	yes	—
215	C₄²	C₂×Q₁₆	1	2	yes	—
216	C₄²	C₂²×Q₈	1	1	yes	characteristic
217	C₄²	C₂×#41	1	1	yes	characteristic
218	C₄²	H(5,𝔽₂)	1	1	yes	characteristic
219	C₄²	2₋¹⁺⁴	1	1	yes	characteristic
220	C₂³	C₂²×C₄⋊C₄	1	1	yes	characteristic
221	Q₈	—	2	8	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	#423	1	2	yes	—
241	C₂×C₄	#423	1	2	yes	—
242	C₂×C₄	C₂×#107	1	2	yes	—
243	C₂×C₄	C₂×#107	1	4	yes	—
244	C₂×C₄	C₂×#107	1	4	yes	—
245	C₂×C₄	C₂×#107	1	2	yes	—
246	C₂×C₄	C₂×#108	1	4	yes	—
247	C₂×C₄	C₂×#108	1	2	yes	—
248	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
249	C₂×C₄	#520	1	1	yes	characteristic
250	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
251	C₂×C₄	#550	1	1	yes	characteristic
252	C₂×C₄	#557	1	1	yes	characteristic
253	C₂×C₄	Q₈²	1	1	yes	characteristic
254	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
255	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
256	C₂×C₄	C₂×#131	1	1	yes	characteristic
257	C₂×C₄	#568	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
259	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
260	C₂×C₄	#576	1	1	yes	characteristic
261	C₂×C₄	#578	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	—	2	2	no	—
264	C₈	—	2	4	no	—
265	C₈	—	4	4	no	—
266	C₈	—	4	2	no	—
267	C₈	—	4	1	no	—
268	C₈	—	8	2	no	—
269	C₈	—	8	4	no	—
270	C₈	#446	1	4	yes	complemented
271	C₂²	Q₈×C₄⋊C₄	1	1	yes	characteristic
272	C₂²	C₂×#423	1	1	yes	characteristic
273	C₂²	#2906	1	1	yes	characteristic
274	C₂²	C₂²×#107	1	1	yes	characteristic
275	C₂²	#2913	1	1	yes	characteristic
276	C₂²	C₂²×#108	1	1	yes	characteristic
277	C₂²	#2916	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	4	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	2	no	—
286	C₄	—	4	1	no	—
287	C₄	#1567	1	4	yes	—
288	C₄	C₂×#446	1	1	yes	characteristic
289	C₄	Q₈×M₁₆	1	1	yes	characteristic
290	C₄	#2989	1	1	yes	characteristic
291	C₄	#2990	1	1	yes	characteristic
292	C₄	Q₈×D₈	1	1	yes	characteristic
293	C₄	Q₈×Q₁₆	1	1	yes	characteristic
294	C₄	#3391	1	1	yes	characteristic
295	C₄	#3395	1	1	yes	characteristic
296	C₂	C₂×#1567	1	1	yes	characteristic
297	C₂	#17799	1	1	yes	characteristic
298	C₂	#17828	1	1	yes	characteristic
299	C₂	#17836	1	1	yes	characteristic
300	C₂	#22722	1	1	yes	characteristic
301	C₂	Q₈×#107	1	1	yes	characteristic
302	C₂	Q₈×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂²×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
81. #335	Q₈	semidirect
270. C₈	#446	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22722	2
C₄×C₈×Q₈	2
#9604	2
#9601	2
#8806	2
#8274	2
C₄×#335	2